Lecture 11 - Attrition
Column (1) shows the average for the group assigned catastrophic coverage. Columns (2)–(5) compare averages in the deductible, cost-sharing, free care, and any insurance groups with the average in column (1).
Source: Angrist and Piscke (2014, 19)
\[r_i = r_i(0)(1 - z_i) + r_i(1)z_i \]
\[Y_i(z) \perp R_i(z)\]
\[Y_i(z) \perp R_i(z) | X_i\]
\[E[Y_i(1)] = \frac{1}{N} \sum_{i=1}^{N} \frac{Y_i(1) r_i(1)}{\pi_i(z = 1, x)},\]
| Observation | \(Y_i(0)\) | \(Y_i(1)\) | \(r_i(0)\) | \(r_i(1)\) | \(Y_i(0) \vert r_i(0)\) | \(Y_i(1) \vert r_i(1)\) | \(X_i\) |
|---|---|---|---|---|---|---|---|
| 1 | 3 | 4 | 1 | 1 | 3 | 4 | 1 |
| 2 | 4 | 7 | 1 | 1 | 4 | 7 | 1 |
| 3 | 3 | 4 | 1 | 1 | 3 | 4 | 1 |
| 4 | 4 | 7 | 1 | 1 | 4 | 7 | 1 |
| 5 | 10 | 14 | 0 | 0 | Missing | Missing | 0 |
| 6 | 12 | 18 | 0 | 0 | Missing | Missing | 0 |
| 7 | 10 | 14 | 1 | 1 | 10 | 14 | 0 |
| 8 | 12 | 18 | 1 | 1 | 12 | 18 | 0 |
\[E[Y_i(1)] = (\frac{1}{8}) (\frac{4}{1} + \frac{7}{1} + \frac{4}{1} + \frac{7}{1} + \frac{14}{0.5} + \frac{18}{0.5}) = 10.75,\]
\[E[Y_i(0)] = (\frac{1}{8}) (\frac{3}{1} + \frac{4}{1} + \frac{3}{1} + \frac{4}{1} + \frac{10}{0.5} + \frac{12}{0.5}) = 7.25,\]
\[E[Y_i(1)] - E[Y_i(0)] = 3.5.\]
Imagine the missing outcomes from those who dropped out
We don’t know what those outcomes are, but we can consider two extreme scenarios:
Best-case scenario: Assume the dropouts in the treatment group would have had the best possible outcomes, and dropouts in the control group would have had the worst possible outcomes (relative to the observed data)
Worst-case scenario: Assume the dropouts in the treatment group would have had the worst possible outcomes, and dropouts in the control group would have had the best possible outcomes
By calculating the ATE under these extreme assumptions, we get a range (upper bound and lower bound) that (hopefully 😅) contains the true ATE
To find the upper bound on the treatment effect estimate, substitute 10 for the missing values in the treatment group and 0 for the missing value in the control group
Upper bound: \(\frac{(7 + 10 + 10 + 10)}{4} - \frac{(0 + 7 + 5 + 6)}{4} = \frac{37}{4} - \frac{18}{4} = 4.75\)
To find the lower bound on the treatment effect estimate, substitute 0 for the missing values in the treatment group and 10 for the missing value in the control group
Lower bound: \(\frac{(7 + 10 + 0 + 0)}{4} - \frac{(10 + 7 + 5 + 6)}{4} = \frac{17}{4} - \frac{28}{4} = -2.75\)
So, the ATE is between -2.75 and 4.75… and indeed it is (2)! 🎉
But as you can see, the bounds are pretty wide…